Pitts monads and a lax descent theorem

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Abstract

A theorem of A.M.Pitts (1986) states that essential surjections of toposes bounded over a base topos S are of effective lax descent. The symmetric monad M on the 2-category of toposes bounded over S is a KZ-monad (Bunge-Carboni 1995) and the M-maps are precisely the S-essential geometric morphisms (Bunge-Funk 2006). These last two results led me to conjecture and then prove the general lax descent theorem that is the subject matter of this paper.By a 'Pitts KZ-monad' on a 2-category K it is meant here a locally fully faithful equivariant KZ-monad M on K that is required to satisfy an analogue of Pitts' theorem on bicomma squares along essential geometric morphisms. The main result of this paper states that, for a Pitts KZ-monad M on a 2-category K ('of spaces'), every surjective M-map is of effective lax descent. There is a dual version of this theorem for a Pitts co-KZ-monad N. These theorems have (known and new) consequences regarding (lax) descent for morphisms of toposes and locales.

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